MathDB

2

4x1, 2 3x1, 3 2x1, 4 1x1 pieces

4\times 1

rectangle, two

3\times 1

rectangles, three

2\times 1

rectangles, and four

1\times 1

squares. Ariel and Bernardo play the following game on a board of

n\times n

, where

n

is a number that Ariel chooses. In each move, Bernardo receives a piece

R

from Ariel. Next, Bernardo analyzes if he can place

R

on the board so that it has no points in common with any of the previously placed pieces (not even a common vertex). If there is such a location for

R

, Bernardo must choose one of them and place

R

. The game stops if it is impossible to place

R

in the way explained, and Bernardo wins. Ariel wins only if all

10

pieces have been placed on the board. a) Suppose Ariel gives Bernardo the pieces in decreasing order of size. What is the smallest n that guarantees Ariel victory? b) For the

n

found in a), if Bernardo receives the pieces in increasing order of size, is Ariel guaranteed victory?
Note: Each piece must cover exactly a number of unit squares on the board equal to its own size. The sides of the pieces can coincide with parts of the edge of the board.

red points on 24 intersections of 2x7 grid

In a

2\times 7

board gridded in

1\times 1

squares, the

24

points that are vertices of the squares are considered. https://cdn.artofproblemsolving.com/attachments/9/e/841f11ef9d6fc27cdbe7c91bab6d52d12180e8.gif Juan and Matías play on this board. Juan paints red the same number of points on each of the three horizontal lines. If Matthias can choose three red dots that are vertices of an acute triangle, Matthias wins the game. What is the maximum number of dots Juan can color in to make sure Matías doesn't win? (For the number found, give an example of coloring that prevents Matías from winning and justify why if the number is greater, Matías can always win.)

3

2

consecutive integers such that sum of their squares is a square

Find the minimum

k>2

for which there are

k

consecutive integers such that the sum of their squares is a square.

coloring positive integers into 3 colors

Is it possible to color positive integers with three colors so that whenever two numbers with different colors are added, the result of their addition is the third color? (All three colors must be used.) If the answer is yes, indicate a possible coloration; if not, explain why.

1

2

min natural with all digits 4, multiple of 169

Determine the smallest positive integer that has all its digits equal to

4

, and is a multiple of

169

.

volume of parallelepiped (May Olympiad 2010 L1 - Olimpiada de Mayo)

A closed container in the shape of a rectangular parallelepiped contains

1

liter of water. If the container rests horizontally on three different sides, the water level is

2

cm,

4

cm and

5

cm. Calculate the volume of the parallelepiped.

2

equal segments starting with a rectangle and a circle

Let

ABCD

be a rectangle and the circle of center

D

and radius

DA

, which cuts the extension of the side

AD

at point

P

. Line

PC

cuts the circle at point

Q

and the extension of the side

AB

at point

R

. Show that

QB = BR

.

find the pattern

In stage

0

the numbers are written:

1 , 1

. In stage

1

the sum of the numbers is inserted:

1, 2, 1

. In stage

2

, between each pair of numbers from the previous stage, the sum of them is inserted:

1, 3, 2, 3, 1

. One more stage:

1, 4, 3, 5, 2, 5, 3, 4, 1

. How many numbers are there in stage

10

? What is the sum of all the numbers in stage

10

?

4

2

Suggestions

Let

n

be a integer

1<n<2010

, where we have a polygon with

2010

sides and

n

coins, we have to paint the vertices of this polygon with

n

colors and we've to put the

n

coins in

n

vertices of the polygon. In each second the coins will go to the neighbour vertex, going in the clockwise. Determine the values of

n

such that is possible paint and choose the initial position of the coins where in each second the

n

coins are in vertices of distinct colors

multiples of 13 wanted

Find all natural numbers of

90

digits that are multiples of

13

and have the first

43

digits equal to each other and nonzero, the last

43

digits equal to each other, and the middle

4

digits are

2, 0, 1, 0

, in that order.

2010 May Olympiad

Subcontests

4x1, 2 3x1, 3 2x1, 4 1x1 pieces

red points on 24 intersections of 2x7 grid

consecutive integers such that sum of their squares is a square

coloring positive integers into 3 colors

min natural with all digits 4, multiple of 169

volume of parallelepiped (May Olympiad 2010 L1 - Olimpiada de Mayo)

equal segments starting with a rectangle and a circle

find the pattern

Suggestions

multiples of 13 wanted